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Let $X_1, . . . , X_n$ be random variables uniformly distributed over the interval $[a, b]$.

Let $Y_1 < Y_2 < · · · < Y_n$ be the same values in sorted order.

Let $Y_0 = a$ and $Y_{n+1} = b$. Show that each random variable $Z_i = Y_{i+1} − Y_i$ has the same distribution. [Hint: show that for $i > 0 : Y_{i+1} − Y_i$ has the same distribution as $Y_i − Y_{i−1}$.]

What I figure out right now is that I could compute the CDF to see if they are the same, but I am getting stuck to find out the CDF, or is there any other way to proof it?

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  • $\begingroup$ Have you learnt about the joint distribution of ordered statistics? $\endgroup$ – BGM Apr 20 '16 at 3:51
  • $\begingroup$ @BGM Sorry, I know the join distribution but not of the ordered statistics. $\endgroup$ – Jason Apr 20 '16 at 3:58
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Conditioned on $Y_{i-1}$ and $Y_{i+1}$, $Y_i$ has uniform distribution on $[Y_{i-1}, Y_{i+1}]$. Thus conditioned on $Y_{i-1}$ and $Y_{i+1}$, both $Y_i - Y_{i-1}$ and $Y_{i+1} - Y_i$ are uniform on $[0, Y_{i+1} - Y_{i-1}]$. Since the conditional distributions given $Y_{i-1}$ and $Y_{i+1}$ are the same, the unconditional distributions are the same.

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